We present results on the last topic we collaborate with our late friend, Professor Ajoy Kumar Datta (1958-2019). In this work, we shed new light on a self-stabilizing wave algorithm proposed by Colette Johnen in 1997. This algorithm constructs a BFS spanning tree in any connected rooted network. Nowadays, it is still the best existing self-stabilizing BFS spanning tree construction in terms of memory requirement,(it only requires Θ(1) bits per edge). However, it has been proven assuming a weakly fair daemon. Moreover, its stabilization time was unknown. Here, we study the slightly modified version of this algorithm, still keeping the same memory requirement. We prove the self-stabilization of this variant under the distributed unfair daemon and show a stabilization time in O(D.n^2) rounds, where D is the network diameter and n the number of processes.